Tangential acceleration of a pendulum

Question

A pendulum mass is dropped from 90 degrees and t=0 is when the mass passes through its lowest point for the first time. Which of the following is a reasonable graph for the tangential acceleration of the pendulum over time?

a) A triangular curve: one that is extra flat as it crosses equilibrium and kind of pointy near the peaks

b) A saw tooth curve: a fairly flat, positive slope is followed by a near vertical drop

c) A square curve: one that is flat near the peaks and fairly vertical as it crosses equilibrium

d) A sawtooth curve: a fairly flat, negative slope is followed by a near vertical rise

Please provide an explanation

Answer 1

Hey Emily! Let's think about the forces acting on the system. There is the downward gravitational force on the pendulum, and there is the radial (towards the pivot point) tension force on the pendulum. This tension force is always perpendicular to the tangential acceleration, so we don't have to consider it for this problem. Only the gravitational force contributes to the tangential acceleration and remember, F = ma so the tangential acceleration is proportional to the tangential component of the gravitational force on the pendulum. So, where is the tangential gravitational force (and thus the tangential acceleration) the strongest? It's at the points of maximum height, or at 90 degrees on either side, where the downward force vector aligns perfectly with the tangent of the pendulum's path. As the pendulum falls, the tangential component of the downward gravity force gets smaller and smaller until the bottom of the path, or the "equilibrium" point (at t=0), where the gravitational force is perpendicular to the path and the instantaneous tangential acceleration is zero. Once it passes equilibrium, the tangential force (and thus the tangential acceleration) gets bigger and bigger until it's turn-around point on the other side, and the cycle repeats itself. For this reason, the answer is A, a triangular curve.

Another way to approach this problem is to think about the symmetry of the physical system. If you drop a pendulum from either side (neglecting any friction), it will swing back and forth with symmetry about the center line of the system. In other words, the mirror image of the system would look the same. For this reason, we can rule out b and c, the sawtooth curves. These graphs are not symmetric about the turnaround point (ie, if you look at a mirror image of these graphs, it would not look the same as the original graph). For this reason, it cannot describe our symmetric system. Next, we consider the fact that the tangential acceleration is proportional to the tangential component of the gravitational force (see explanation in prev. paragraph), which we know decreases gradually, or "continuously," as the pendulum swings down, and then increases continuously as it swings back up. For this reason, we can rule out the square wave, because a square wave would indicate sharp, discontinuous changes in direction, which does not properly describe the system. Thus, we are left with A.